How to test if a curve is well described by an ellipse?

I have a set of data points in 2D, and I am trying to come up with some sort of statistical to determine if the points fall along an ellipse. My idea so far is to fit an ellipse to the points, take the main square error, and use this as an indicator. However, this requires me to set some threshold for what is a good MSE (so MSE above this threshold indicate that the points do not fall along an ellipse). Might there be a better way to go about this?